Blow-Up Phenomena for a Class of Parabolic Systems with Time Dependent Coefficients ()
1. Introduction
It is well known that the solutions of parabolic problems may remain bounded for all time, or may blow-up in finite or infinite time. When blow-up occurs at time
, the evaluation of
is of great practical interest.
In a recent paper [1] Payne and Schaefer have investigated the blow-up phenomena of solutions in some parabolic systems of equations under homogeneous Dirichlet boundary conditions. The contribution of this note is to extend their investigations to a class of parabolic systems with time dependent coefficients. The case of a single parabolic equation was investigated recently in [2].
There is an abounding literature dealing with blow-up phenomena of solutions to parabolic partial differential equations. We refer the interested readers to [3-5]. A variety of physical, chemical, biological applications are discussed in [5,6]. Further references to the field are [1,7-19]. In this note we investigate the blow-up phenomena of the solution
of the following parabolic system
(1.1)
where
is a bounded domain in
. The initial data
as well as the data
are assumed nonnegative, so that the solution
of (1.1) will be nonnegative by the maximum principle. More specific assumptions on the data will be made later.
In Section 2 we derive conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and derive under these conditions some upper bound for
. In Section 3 we derive some lower bounds for the blow-up time
when blow-up occurs. However this section is limited to the case of
in
and in
respectively, because our technique makes use of some Sobolev type inequalities available in
and in
only. For convenience we include the proof of one of these inequalities in Section 4.
2. Conditions for Blow-Up in Finite Time t*
Let
be the first eigenvalue and
be the associated eigenfunction of the Dirichlet-Laplace operator defined as
(2.1)
(2.2)
Let the auxiliary function
be defined in
as
(2.3)
with
(2.4)
where
is the solution of problem (1.1). We assume in this section that
is a bounded domain of
, and that
(2.5)
(2.6)
We then compute
(2.7)
Making use of Hölder’s inequality, we have
(2.8)
Combining (2.7) and (2.8), we obtain
(2.9)
A similar computation leads to
(2.10)
Adding (2.9) and (2.10), we obtain
(2.11)
where
is defined in (2.6). We first investigate the particular case
. Making use of Hölder’s inequality, we have
(2.12)
Inserted in (2.11), we obtain the first order differential inequality
(2.13)
Integrating (2.13) from 0 to
, we obtain the inequality
(2.14)
Suppose that the data satisfy the condition
(2.15)
Then
vanishes at some time
, and
must blow up at some time
. We obtain
(2.16)
In the general case, we suppose without loss of generality that
, and make use of the inequality
(2.17)
valid for arbitrary
. Choosing
, we obtain
(2.18)
with
(2.19)
Inserted in (2.12), we obtain the first order differential inequality
(2.20)
Suppose that the initial data are so large that
. Then
is increasing for t small. Since
is increasing in
from its negative minimum, it follows then that
is increasing for
. This shows that
remains positive, so that
blows up at time
. Integrating (2.20) leads to the following upper bound for ![](https://www.scirp.org/html/4-7400752\7c3befde-3fdd-458f-9e61-887b36d81259.jpg)
(2.21)
These results are summarized in the following.
Theorem 1
1) Assume (2.5) with
, (2.6), and (2.15). Then
defined in (2.3) blows up at finite time
bounded above by (2.16).
2) Assume (2.5) with
, (2.6), and
with
defined in (2.20). Then
blows up at finite time
bounded above by (2.21).
To conclude this section, we note that if the condition (2.6) is replaced by
(2.22)
then we have to replace the initial data
by
in Theorem 1. Clearly we may use a lower bound for
. For instance we may integrate the differential inequality
(2.23)
that follows from (2.11), leading to the lower bound
(2.24)
3. Lower Bounds for t*
In this section we assume that the data
satisfy the conditions
(3.1)
and that the data
are nonnegative for all
. Moreover the solution is assumed to blow up in the sense that
as
, where
is defined as
(3.2)
with
(3.3)
(3.4)
Differentiating (3.3) and making use of (1.1), (3.1), we obtain
(3.5)
with
(3.6)
Making use of Schwarz and Hölder’s inequalities we have
(3.7)
In
we make use of the following Sobolev type inequality
(3.8)
derived in the last section of the paper. Combining (3.7) and (3.8), we obtain
(3.9)
where we have used the arithmetic-geometric mean inequality. Making use of the inequality
(3.10)
we have
(3.11)
valid for arbitrary
to be chosen later. Inserted in (3.9) and (3.5), we obtain
(3.12)
We now select
(3.13)
in order to have
in (3.12), arriving at
(3.14)
with
(3.15)
A similar computation leads to
(3.16)
where
is defined in (3.4). In
, we replace (3.7) by
(3.17)
and make use of the Sobolev type inequality
(3.18)
derived by Talenti in [20] with
. Inserted in (3.17), we obtain
(3.19)
with
(3.20)
Moreover we make use of (3.10) to write
(3.21)
with arbitrary
to be chosen later. Combining (3.5), (3.19) and (3.21), we obtain
![](https://www.scirp.org/html/4-7400752\3ec99e3d-dd2c-4e52-88a6-684df2190315.jpg)
We now select
such that the quantity
in (3.22) vanishes. We are then led to the inequality
(3.23)
with
(3.24)
Finally we make use of (3.10) to write
(3.25)
and select
to satisfy
, leading to
(3.26)
Inserted in (3.23), we obtain
(3.27)
with
(3.28)
A similar computation leads to
(3.29)
If we suppose that
(3.30)
then there exists
such that
and we have
(3.31)
valid for
, with
(3.32)
(3.33)
(3.34)
Integrating (3.31), we obtain in the two-dimensional case
(3.35)
from which we obtain a lower bound for
of the form
(3.36)
where
is the inverse function of
. In the threedimensional case, we obtain
(3.37)
from which we obtain a lower bound for
of the form
(3.38)
These results are summarized in the following
Theorem 2
Under the assumption (3.30), a lower bound for the blow-up time t* of the solution
of (1.1) is given by (3.36) in the two-dimensional case and by (3.38) in the three-dimensional case.
In the particular case in which
and
are constant, we have
(3.39)
in the two-dimensional case and
(3.40)
in the three-dimensional case.
Theorem 2 could easily be extended to systems of n parabolic equations of the form
(3.41)
4. Sobolev Type Inequality in ![](https://www.scirp.org/html/4-7400752\c75db69c-6739-4902-b9bd-9447e5bad619.jpg)
The Sobolev type inequality (3.8) in
may be known, but for the convenience of the reader we present a proof here.
Lemma 1
Let
be a nonnegative piecewise
-function defined in a bounded domain
that vanishes on the boundary
. Let
be any constant
. Then we have the following Sobolev type inequality
(4.1)
valid for
.
For the proof of (4.1), we follow the argument of Payne in [21]. We note that (4.1) is equivalent to
(4.2)
where
is the convex hull of
, and
It is therefore sufficient to establish (4.1) for
convex. For the proof, let
be an arbitrary point in
Let
be two pairs of boundary points associated to P with![](https://www.scirp.org/html/4-7400752\9f8a2e66-3062-4951-b462-cb71e37f9383.jpg)
. Since
vanishes on
, we have for any constant ![](https://www.scirp.org/html/4-7400752\f19a2b2d-ab9d-4de0-b815-221fe7d0c471.jpg)
(4.3)
from which we obtain
(4.4)
Similarly we have
(4.5)
Multiplying (4.4) by (4.5) and integrating over
leads to
![](https://www.scirp.org/html/4-7400752\2e4a749f-76d5-4b76-9d9a-6f1e09e8b4d3.jpg)
(4.6)
which is the desired inequality (4.1). We note that we have used the Schwarz and the arithmetic-geometric mean inequalities in the two last steps of (4.6).